Existence of function
Posted: Sat Jan 16, 2016 2:04 am
Let \(\displaystyle{x\,,y:\left[0,1\right]\longrightarrow \mathbb{R}\,\, t\mapsto x(t)\,,t\mapsto y(t)}\) be \(\displaystyle{\rm{C}^{\infty}}\) functions
with \(\displaystyle{\left(x(t),y(t)\right)\neq \overline{0}\,\,,\forall\,t\in\left[0,1\right]}\) .
Consider the functions \(\displaystyle{a\,,b:\left[0,1\right]\longrightarrow \mathbb{R}}\,,a(t)=\dfrac{x(t)}{\sqrt{x^2(t)+y^2(t)}}\,,b(t)=\dfrac{y(t)}{\sqrt{x^2(t)+y^2(t)}}\)
such that \(\displaystyle{\cos\,\phi_{0}=a(0)\,,\sin\,\phi_{0}=b(0)}\) for some \(\displaystyle{\phi_{0}\in\mathbb{R}}\) .
Prove that there exists a function \(\displaystyle{\phi:\left[0,1\right]\longrightarrow \mathbb{R}}\) such that
\(\displaystyle{\cos\,\phi(t)=a(t)\,,\sin\,\phi(t)=b(t)\,,\forall\,t\in\left[0,1\right]}\) .
with \(\displaystyle{\left(x(t),y(t)\right)\neq \overline{0}\,\,,\forall\,t\in\left[0,1\right]}\) .
Consider the functions \(\displaystyle{a\,,b:\left[0,1\right]\longrightarrow \mathbb{R}}\,,a(t)=\dfrac{x(t)}{\sqrt{x^2(t)+y^2(t)}}\,,b(t)=\dfrac{y(t)}{\sqrt{x^2(t)+y^2(t)}}\)
such that \(\displaystyle{\cos\,\phi_{0}=a(0)\,,\sin\,\phi_{0}=b(0)}\) for some \(\displaystyle{\phi_{0}\in\mathbb{R}}\) .
Prove that there exists a function \(\displaystyle{\phi:\left[0,1\right]\longrightarrow \mathbb{R}}\) such that
\(\displaystyle{\cos\,\phi(t)=a(t)\,,\sin\,\phi(t)=b(t)\,,\forall\,t\in\left[0,1\right]}\) .